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American Journal of Physics

Evolution of a Gaussian Wavepacket

S. M. Blinder

Published Jun 1, 1968• 22 citations
Publisher

Abstract

An alternative derivation is given for the evolution in time of a Gaussian wavepacket representing a free particle. This is based on explicit evaluation of the action of the evolution operator on the initial wavefunction, rather than on expansions in free-particle eigenstates. Both the cases of a free particle initially at rest and in uniform linear motion are considered.

DOI
10.1119/1.1974961

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The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.

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Analysis of the spreading Gaussian wavepacket

Géza I. Márk

The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)

European Journal of Physics•May 1, 1997•32 citations
An alternative derivation is given for the evolution in time of a Gaussian wavepacket representing a free particle. This is based on explicit evaluation of the action of the evolution operator on the initial wavefunction, rather than on expansions in free-particle eigenstates. Both the cases of a free particle initially at rest and in uniform linear motion are considered.
The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.
Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.
This chapter focuses on a basic but systematic treatment of wavefunctions. By comparison with a spin 1/2 system, the concept of a wavefunction as a collection of coefficients is developed, followed by discussions of eigenstates, operators, and representations in the context of wavefunctions. Transformation from position to momentum representation is effected by a Fourier transform. This is developed next, and illustrated for a Gaussian. The discussion of phases and time evolution in earlier chapters is now applied to wavefunctions, and the time evolution of a free particle discussed. The concept of a wavepacket and its suitability for representing particles follows. A topic of some subtlety — and one which is discussed in detail — is the use of bra-ket (Dirac) notation for wavefunctions. The chapter closes with a short epilogue.
We present a simple approach for finding the general solution of the time-dependent Schrödinger equation for a particle moving under the action of a uniform force. From such a general solution we derive the eigenfunctions of the energy as well as those of the momentum. Next we obtain the propagator and show that the time evolution of an arbitrarily chosen initial wavefunction is described by a Fresnel transformation. This establishes a useful link with optical phenomena and allows us to link the case of a particle moving under a uniform force to that of the free particle.
The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)